High throughput high resolution gas sorption screening

ABSTRACT

A system and method for high-throughput, high-resolution gas sorption screening are provided. An example system includes a sample chamber with a hermetic seal and a heat exchanger system. The heat exchanger system includes a heat exchanger disposed in the sample chamber, a coolant circulator fluidically coupled to the heat exchanger, and a sample plate comprising sample wells in contact with the cooling fluid from the coolant circulator. The system also includes a gas delivery system. The gas delivery system includes a gas source and a flow regulator. A temperature measurement system is configured to sense the temperature of the sample wells.

TECHNICAL FIELD

The present disclosure is directed to screening of sorbents forefficacy.

BACKGROUND

Porous materials, such as zeolites, metal-organic frameworks (MOFs),covalent-organic frameworks (COFs), and activated carbons are used ingas separation and storage applications. Fast high-throughput screeningtools are needed to characterize and screen the sorption performance ofa large number of these materials for particular applications. Suchtools would allow for the exhaustive search for an optimized sorbentfrom a pool of thousands. Furthermore, high-throughput screening toolscould improve materials discovery by allowing the development ofstructure-performance relationships.

Most high-throughput screening methods rely on computer simulations ofthe adsorption phenomenon. The accuracy of the computational methodsdepends on accurate knowledge of the structure of the adsorbents and theadsorbate-adsorbent interactions. In contrast, experimental screeningmethods would allow for the characterization of materials with unknownstructures and surface chemistry. The success of developing a reliablelaboratory high-throughput screening equipment depends on relating aneasily measurable quantity, such as the sample's temperature orpressure, with the material's adsorption capacity.

SUMMARY

An embodiment described herein provides a system for screening sorbents.The system includes a sample chamber with a hermetic seal and a heatexchanger system. The heat exchanger system includes a heat exchangerdisposed in the sample chamber, a coolant circulator fluidically coupledto the heat exchanger, and a sample plate comprising sample wells incontact with the cooling fluid from the coolant circulator. The systemalso includes a gas delivery system. The gas delivery system includes agas source and a flow regulator. A temperature measurement system isconfigured to sense the temperature of the sample wells.

An embodiment described herein provides a system for screening sorbents.The system includes a sample chamber with a hermetic seal and a heatexchanger system. The heat exchanger system includes a heat exchangerdisposed in the sample chamber, a coolant circulator fluidically coupledto the heat exchanger, and a sample plate comprising sample wells incontact with the cooling fluid from the coolant circulator. The systemalso includes a gas delivery system. The gas delivery system includes agas source and a flow regulator. A temperature measurement system isconfigured to sense the temperature of the sample wells.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic drawing of a system for high-throughput,high-resolution, gas sorption screening of sorbents.

FIG. 2 is a flowchart of a method for using the apparatus to screensorbents.

FIG. 3A is a plot of methane adsorption isotherms of a set of selectedmaterials at 10° C.

FIG. 3B is a plot of the transient temperature profile of the selectedmaterials for simulation case I.

FIG. 4A is a plot of the generated heat profile when the adsorptionexperiment is performed using a cooling fluid with a temperature of 10°C.

FIG. 4B is a plot of the calculated value of the generated heat ofadsorption of the selected materials over time.

FIG. 5 is a plot of the calculated heat of adsorption of the selectedmaterials using Equation 5.11.

FIGS. 6A and 6B are plots of estimated adsorption isotherms for theselected materials and a comparison with actual values.

FIG. 7 is a plot of model adsorption isotherms of the selectedmaterials.

FIG. 8 is a plot of temperature profiles during adsorption of CO₂.

FIG. 9 is a plot of temperature profiles during adsorption of CO₂.

FIGS. 10A and 10B are plots of the estimated adsorption isotherms forselected materials and their comparisons with actual values.

DETAILED DESCRIPTION

Methods and systems described herein provide a high-throughput screeningtechnique that can assess the sorption performance of different classesof porous materials. The disclosed technologies link temperaturechanges, measured using an infrared camera, during the adsorptionprocess with the materials' sorption uptake. A test gas is continuouslyfed to a test chamber at a controlled pressurization rate. Thedissipation of heat generated by adsorption is controlled using a shelland tube heat exchanger-like configuration. The continuous flow of thetest gas into the sample chamber enables the generation ofhigh-resolution adsorption isotherms (HRI) allowing materials to beranked at various operating conditions. Control of the heat dissipationrate permits the use of novel formulas that relate the amount of gasthat sort at equilibrium with the observed temperature changes. Theformulas are independent of the materials' thermo-physical propertiesallowing the testing and comparison of materials with differentproperties. Derivation of these formulas and their application togenerate the FRI are described.

The apparatus design and screening methods account for variations in thethermal conductivity and heat capacity between the different sorbents inorder to minimize their contribution to the apparent changes intemperature during the sorption process. The continuous flow of gas tothe chamber in this design allows the determination of HRI in a shortertime than other techniques. Accordingly, the device's lower sensitivityto the isosteric heat of adsorption permits the development of ageneralized method to screen different classes of materials withappreciable differences in their thermo-physical properties.Accordingly, the performance of different classes of porous materialssuch as MOFs, zeolites, and activated carbons can be screenedsimultaneously. The HRIs obtained allow for comparative studies atvarious operating conditions.

FIG. 1 is a schematic drawing of a system 100 for high-throughput,high-resolution, gas sorption screening. The system proposed in thisinvention includes a sample chamber 102, which is designed to behermetically sealed. A heat exchanger system is used to control thetemperature of the samples, and includes a heat exchanger 104 and acoolant circulator 106. It can be noted that the coolant circulator 106may be used to raise, lower, or hold the temperature of the heatexchanger 104. A gas delivery system provides the test gas or test gasmixture, and includes a gas source 108 and a flow controller 110. Atemperature measurement system, such as an infrared camera is used tosense the temperature of the samples during adsorption. A pressurecontrol system is used to pull a vacuum on the sample chamber 102 and torelease pressure once the experiment is complete. The venting systemincludes a vacuum pump 114 and a vent valve 116. A control system 118,such as a personal computer or laboratory server, is used to monitor thetemperature over time through the temperature monitoring system 112,control the gas flow through the flow controller 110, and monitor thepressure through a pressure sensor 120.

In one embodiment, the gas delivery system comprises a test gas cylinderas the gas source 108 connected to the flow controller 110, upstream thesample chamber 102. The flow controller 110 is calibrated to control theflow rate or the pressurization rate of the sample chamber 102. Inanother embodiment, a pressure controller, for example, as part of theflow controller 110 is used to control the pressurization rate. Gassorption takes place in the heat exchanger 104, which is placed insidethe sample chamber 102. The tubes are open from the top side, forming asample plate 122, and are sealed from the bottom creating sample wells123. The sample wells 123 should have a high aspect ratio to reducethermal gradient in the axial direction, and a radius less than 2 mm toreduce the material thermal resistance and enhance heat dissipation. Thewall thickness should be same for all sample wells 123 to account forits effect on the overall heat transfer coefficient. The candidatematerials are loaded into the sample wells 123 being exposed from thetop to the test gas. The heat generated by the adsorption is transferredto the cooling fluid flowing in the shell side of the heat exchanger104. In the sample chamber 102, the test gas is introduced from the topthrough a nozzle 124 to enhance uniform distribution of the test gas.The sample chamber 102 is sealed to maintain the desired internalpressure and environment.

The sample chamber 102 can be made of any material that can withstandthe operating conditions such as stainless steel, titanium, or ceramic,among others. The lid 126 of the sample chamber 102 is generally madefrom the same material as the sample chamber 102, and has aninfrared-transparent window in the center, above which the temperaturesensor 112 is mounted to monitor the temperature change of the samplewells 123 at known time intervals. As used herein, aninfrared-transparent window will allow a subset of infrared lightwavelengths to pass through at a sufficient level to allow thetemperatures of sample wells 123 to be measured, for example, at atransmission of 30% or higher. In various embodiments, theinfrared-transparent window is quartz or polycarbonate, among others.The wavelengths depend on the measurement and the window selected. Forexample, a quartz window may transmit infrared wavelengths between 0.2μm and about 3.0 μm. A polycarbonate window may transmit infraredwavelengths between 0.5 μm and 1000 μm. In some embodiments, sufficienttransmission, e.g., greater than about 30%, is achieved at wavelengthsof interest, e.g., between about 1.5 μm and about 9 μm using a glasswindow, such as a chalconide infrared glass. Other materials, such asfiberglass, may be used in embodiments.

In some embodiments, the temperature sensor 112 is an infrared camera,such as a forward-looking infrared (FLIR) camera. The accuracy of theadsorption capacity estimate depends on the resolution of thetemperature sensor 112, i.e. a high accuracy and a high temperatureresolution is recommend. In some embodiments, the temperature sensor 112collects data at 60 frames per second or higher. The speed of the datacollection will dictate the resolution of the isotherm obtained for theadsorption.

The distance from the sample plate 122 to the temperature sensor 112position is determined by the size of the sample plate 122, for example,using field-of-view calculators. The heat exchanger 104 is made of aconductive material, such as stainless steel, and may be coated with adark opaque material to minimize back-reflection into the lens of thetemperature sensor 112. The sampled materials are placed in the samplewells 123 of the sample plate 122. The sample wells 123 are arranged inthe sample plate 122 in a staggered configuration to achieve a uniformtemperature profile across the different sample wells 123 upon cooling.The cooling fluid is supplied from the coolant circulator 106 at acontrolled rate and temperature. The cooling fluid is introduced intothe heat exchanger 104 from nozzle 128 and exits from the adjacentnozzle 130. The diameter of the sample wells 123 is less than 2 mm tominimize the effects of thermal conductivity variations between thetested porous materials on the temperature change upon the adsorptionand desorption process. In some embodiments, the sample wells 123 areabout 30 mm in depth.

The sample chamber 102 is vented through the vent valve 116 to allow anymaterials from previous tests to escape, and allow the lid 126 to beremoved. Once the sorbent samples are loaded into the sample wells 123of the sample plate 122, a vacuum pump 114 is used to remove excessgases and adsorbed materials. During this period, the temperature of thesamples may be raised to force any adsorbed materials from the sorbents.

During the test operations, temperature and pressure data is collectedon the control system 118 at pre-defined time intervals, such as every10 seconds, every 5 seconds, every second, or at shorter intervals, suchas 30 frames per second, or 60 frames per second. In some embodiments,the control system 118 is used to automate administration of the testgas, the degassing step, and the operations of the vacuum pump 114, thecoolant circulator 106, and the vent valve 116.

Device Operation Procedure

Sorbents to be screened and characterized for the subject test gas areloaded into the sample wells 123 of the sample plate 122. In someembodiments, some of the sample wells 123 are left empty to use as areference for the temperature measurement. To evacuate the samplechamber 102 of any gases, the isolation valve 132 is opened and thevacuum pump 114 is operated overnight, or at least about 4 hours, atleast about 8 hours, at least about 16 hours, or longer. The chamberpressure is held low, e.g., less than about 20 mm/Hg, or less than about10 mm/Hg, or less than about 5 mm/Hg, or lower, to pull gas residuesfrom the pores of the sample sorbents.

The heat management system is operated at the screening/characterizationtemperature of interest by introducing a temperature modifying fluid, orcoolant, from the coolant circulator 106 through entry nozzle 128. Afterthe temperature of the samples reaches the set point, the velocity ofthe working fluid is fixed and is maintained at the set value throughoutthe sorption experiment. Then, test gas is administered using at a fixedpressurization rate from the gas source 108. The pressurization rateshould be small enough such that the increase in the sample'stemperature is much smaller than the sample's initial temperature.Temperature and pressure are monitored using the temperature sensor 112and pressure sensor 120, respectively. Data is recorded in the controlsystem 118 periodically at known time intervals. Once the experiment isfinish, the heat exchanger system is turned off and the vent valve 116is opened to vent the test gas

Screening Method

To account for the different physical and thermal properties of thetested materials that limit the applicability of published screeningmethods, a novel screening method is outlined herewith. The screeningprocedure is based on exploiting the relationship between temperatureincrease during sorption processes and the adsorption capacity of testedmaterials. Unlike previous methods, the dependence of the observedtemperature increase during the sorption process is derived frommodeling of the transient non-isothermal adsorption dynamics of aparticular sample well, i, within the device chamber shown in FIG. 1 .Based on the derived relationship, conditions at which the sorptionexperiment shall be performed to estimate adsorption isotherms aredescribed.

During gas sorption, the main physical processes that induce changes inthe sample temperature are: 1) heat generated due to gas adsorption (orheat absorbed for endothermic chemisorption processes), 2) heatgenerated due to compression of free gas within the sample void, and 3)heat dissipation/ingress due to the thermal interaction between thesample and the temperature modifying fluid. Variations of the spatiallyaveraged temperature within the sample well assuming that the free gasbehaves as an ideal gas can be written as:

$\begin{matrix}{{\left\lbrack {{\left( {1 - \epsilon} \right)\rho_{s}^{i}C_{p_{s}}^{i}} + {{\epsilon\rho}_{g}C_{p_{g}}}} \right\rbrack\frac{d\;{\Delta(T)}_{i}}{d\; t}} = {{\Delta\; H^{i}\frac{{d(n)}_{i}}{d\; t}} + {\epsilon\frac{d\; p}{d\; t}} - {\frac{U^{i}}{R}\Delta{\left\langle T \right\rangle_{i}.}}}} & (4.1)\end{matrix}$

Here, p is the pressure inside the chamber and R is the radius of thesample wells 123.

n

_(i) is the average adsorbed amount in sample, i, per volume of themedium. Δ

T

_(i)=−T_(f)) where

T

_(i) is the average temperature of sample well i and T_(f) is thetemperature of the cooling fluid. All variables in brackets are volumeaveraged, i.e.

$\left\langle x \right\rangle = {{\frac{1}{R^{2}L}{\int_{0}^{R}{\int_{0}^{L}{x\; d\; z\; r\; d\; r\mspace{14mu}{where}\mspace{14mu} k_{eff}^{i}}}}} = \left\lbrack {{\left( {1 - \epsilon^{i}} \right)k_{s}^{i}} + {\epsilon^{i}k_{g}}} \right\rbrack}$L is the well depth. If the aspect ratio of the sample well, L/R, ismuch larger 1, thermal gradient in the axial direction can be ignoredand thus

$\left\langle x \right\rangle = {\frac{1}{R^{2}}{\int_{0}^{R}{{xrdr} \cdot \rho_{s}^{i}}}}$is the particle density and C_(P) _(s) ^(i) is the material's specificheat capacity, respectively. ΔH^(i) is the isoteric heat of adsorption,which is assumed to be independent of coverage. U^(i) is the overallheat transfer coefficient given by

$\frac{1}{U^{i}} = {\frac{R}{2k_{eff}^{i}} + \frac{1}{h_{f}} + {\frac{\lambda}{k_{wall}}.}}$Here, where ∈^(i) is the porosity of sample i, and k_(s) ^(i) and k_(g)are the thermal conductivity of the solid absorbent and the adsorbatemolecules in the free gas phase, respectively. λ is the wall thicknessof the sample wells 123 and h_(f) is the convective heat transfercoefficient of the cooling fluid that is mainly a function of thecooling fluid Reynolds and Prandtl numbers. For all interior sampleswithin the staggered grid, variations in the value of h_(f) isnegligible. By varying the cooling fluid velocity, one can inducechanges in the values of the heat transfer coefficient and as will bediscussed next can allow for the measurement of the adsorbed amountwithout the need for identifying the specific heat capacity of theadsorbent materials. For instance, to achieve a value of h_(f)=75W/(m²·K) using water as the coolant fluid requires an average flow rateof 20-100 μL/min for wells with a radius of 0.6 mm and sample separationdistance of 1.5 to 3.6 mm. Finally, for endothermic chemisorptionprocesses, the signs of the heat generation and the heat dissipation inEquation (4.5) are reversed.

A mass balance over the device's chamber yields:

$\begin{matrix}{{{V_{void}\frac{d\left\langle c \right\rangle}{d\; t}} = {f - {2\; R^{2}L{\sum\limits_{i = 1}^{N}\frac{d\left\langle n \right\rangle_{i}}{d\; t}}}}},} & (4.2)\end{matrix}$where

c

is the molar concentration of the introduced gas, V_(void) is the voidvolume in the device accounting for the volume of the chamber and thevoids between adsorbent particles and their pores. After loading allsamples, the value of V_(void) can be measured using a helium experimenttypically used to determine the skeletal density of porous media. {dotover (f)} is the molar flow rate to the device that is controlled by theflow controller 105 shown in FIG. 2.1 . If the pressurization rate iscontrolled, Equation 4.2 can be used to calculate the flow rate to thechamber.

To complete the model, one needs an expression for the rate ofadsorption, d

n

t/dt. For most microporous materials, the adsorption rate is given bythe linear driving force approximation:

$\begin{matrix}{\frac{d\left\langle n \right\rangle i}{d\; t} = {{k_{0}^{i}\left( {\left\langle q \right\rangle_{i} - \left\langle n \right\rangle_{i}} \right)}.}} & (4.3)\end{matrix}$Here,

q

_(i) is the averaged adsorbed amount at equilibrium (adsorptionisotherm) and k₀ ^(i) is the mass transfer coefficient that is afunction of all resistances for mass transfer from the gas phase to thepores inside the adsorbent particle. Finally, Equations 4.1-4.3 aresubject to the following initial conditions:

T

(0)=T_(i)=T_(f), p(0)=0,

q

(0)=0, and

n

(0)=0. These conditions can be achieved by circulating the cooling fluidand vacuuming the sample chamber before introducing the gas.

If the sample interiors' Biot number, B_(i) _(s) ≡R_(i) h_(f)/k_(eff)^(i)<<1, and the wall's Biot numbers, b_(l) _(w) ≡h_(f)λ/k_(wall)<<1,the heat transfer coefficient is approximately equal to the convectiveheat transfer coefficient of the cooling fluid and is independent of thethermal conductivity of the adsorbent materials. Thus, the heatdissipation rate for all samples can be assumed to be uniform. Theresistance of heat transfer within the sample wells 123 can be minimizedby reducing their radii. For example, most MOF materials' thermalconductivity is in the range of 0.2-0.5 W/(m K). Thus, the well sampleradius is set to be less than 2 mm, which is sufficient to ignore thethermal resistance of the adsorbents. Since the heat dissipation rate isuniform for all materials, changes in temperature in the differentsamples can be related to the heat of adsorption if one knows the valuesof the material's specific heat capacity. If such information is notknown a priori, one can perform the adsorption experiment at the sametemperature for different cooling fluid velocities and relate theadsorption rate to changes in the observed temperature. To derive suchrelation, Equation (4.1) is integrated over time to obtain:

$\begin{matrix}{{\left\lbrack {{\left( {1 - \epsilon} \right)\rho_{s}^{i}C_{p_{s}}^{i}} + {{\epsilon\rho}_{g}C_{p_{g}}}} \right\rbrack\;\Delta\left\langle T \right\rangle_{i}} = {{{- \frac{h_{f}}{R}}{\int_{0}^{t_{f}}{\Delta\left\langle T \right\rangle_{i}d\; t}}} + {\Delta\; H^{i}\left\langle n \right\rangle_{i}} + {\epsilon\;{p.}}}} & (4.4)\end{matrix}$

At low pressure, the adsorption isotherm can be linearized such that

${\left\langle q \right\rangle \approx {{\frac{\partial\left\langle q \right\rangle}{\partial T}\Delta\left\langle T \right\rangle} + {\frac{\partial\left\langle q \right\rangle}{\partial p}p}}},{{where}\mspace{14mu}\frac{\partial\left\langle q \right\rangle}{\partial T}\mspace{14mu}{and}\mspace{14mu}\frac{\partial\left\langle q \right\rangle}{\partial p}}$are evaluated at the initial conditions. As the different adsorptionisotherms, for most adsorbent materials, converge to 0 as p→0, one canargue that in the low-pressure regime,

${{\frac{{\partial\left\langle q \right\rangle}/{\partial T}}{{\partial\left\langle q \right\rangle}/{\partial p}}\frac{\Delta\left\langle T \right\rangle}{p}}} ⪡ 1.$That is, the adsorbed amount, under low-pressure conditions, can beassumed to be independent of the system's temperature. A similarconclusion of negligible temperature variation effects on

n

_(i) can be drawn from Equation (4.3). Since the heat generated term,ΔH^(i)

n

_(i) in Equation 4.4, is independent of temperature, it can be shownthat:

$\begin{matrix}{{{\Delta\; H^{i}\left\langle n \right\rangle_{i}} = {{\frac{1}{R}\left\{ {{h_{f}^{1}{\int_{0}^{p}{\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{\left( \frac{d\; p}{d\; t} \right)}d\; p}}} + {\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{{\Delta\left\langle T \right\rangle_{i_{2}}\left( p_{s} \right)} - {\Delta\left\langle T \right\rangle_{i_{2}}\left( p_{s} \right)}}{{\quad\quad}\left\lbrack {{h_{f}^{2}{\int_{0}^{p_{2}}{\frac{\Delta\left\langle T \right\rangle_{i_{2}}}{\left( \frac{d\; p}{d\; t} \right)}d\; p}}} - {h_{f}^{1}{\int_{0}^{p_{s}}{\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{\left( \frac{d\; p}{d\; t} \right)}d\; p}}}} \right\rbrack}}} \right\}} - {\epsilon\; p}}},} & (4.5)\end{matrix}$where the subscript in temperature indicates the experimentidentification number. For example, Δ

T

_(i1)(p_(s)) is the temperature variation in experiment 1, i.e. using afluid velocity such as the heat transfer coefficient is equal to h_(f)¹, evaluated at p=p_(s). Here, p_(s) is an arbitrary pressure value ofchoice and it should be small enough such that

${{\frac{{\partial\left\langle n \right\rangle}/{\partial T}}{{\partial\left\langle n \right\rangle}/{\partial p}}\frac{\Delta\left\langle T \right\rangle}{p}}} ⪡ 1$still holds. For most adsorbents, a value of p_(s) less than 1.0 bar isenough. The pressurization rate, dp/dt, can be calculated from themeasured pressure profile. In another embodiment, the pressurizationrate is controlled and is set throughout the experiment. The effectiveheat capacity of the sample, in Equation 4.4, is estimated by:

$\begin{matrix}{\left\lbrack {{\left( {1 - \epsilon} \right)\rho_{s}^{i}C_{p_{s}}^{i}} + {{\epsilon\rho}_{g}C_{p_{g}}}} \right\rbrack = {\frac{1}{{\Delta\left\langle T \right\rangle_{i_{1}}\left( p_{s} \right)} - {\Delta\left\langle T \right\rangle_{i_{2}}\left( p_{s} \right)}}{{\frac{1}{R}\left\lbrack {{h_{f}^{2}{\int_{0}^{p_{2}}{\frac{\Delta\left\langle T \right\rangle_{i_{2}}}{\left( \frac{d\; p}{d\; t} \right)}d\; p}}} - {h_{f}^{1}{\int_{0}^{p_{s}}{\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{\left( \frac{d\; p}{d\; t} \right)}d\; p}}}} \right\rbrack}.}}} & (4.6)\end{matrix}$

To measure the mass transfer resistance of all adsorbents and thusevaluate the equilibrium adsorbed amount,

q

_(i), one can fit the asymptotic solution for

n

_(i) in the region where p→0. At low pressure, the adsorption isothermsof most materials is linear and thus

q

_(i)≈C_(i)t where C_(i)=(∂

q

dp)/(∂p dt) and dp/dt is constant if the pressurization rate is set. Ifthe flow rate is fixed instead, the pressure profile is linearized andthe value of dp/dt, used to calculate C_(i), is equal to thepressurization rate at the beginning of the experiment if the flow rateis fixed. Solving Equation (4.3), using this linear adsorption isotherm,yields:

$\begin{matrix}{\left\langle n \right\rangle_{i} = {{\frac{C_{i}}{k_{0}^{i}}\left\lbrack {{k_{0}^{i}t} + e^{{- k_{0}^{i}}t} - 1} \right\rbrack}.}} & (4.7)\end{matrix}$As t→0, one can linearize the exponential term and show that:

$\begin{matrix}{{\Delta\; H^{i}\left\langle n \right\rangle_{i}} = {{{\frac{\Delta\; H^{i}C_{i}k_{0}^{i}}{2}t^{2}} - {\frac{\Delta\; H^{i}C_{i}{k_{0}^{i}}^{2}}{6}t^{3}} + {O\left( t^{4} \right)}} = {{A_{i}t^{2}} - {B_{i}{t^{3}.}}}}} & (4.8)\end{matrix}$Thus, by fitting the values of ΔH^(i)

n

_(i) obtained from Equation (4.5) with the polynomial described inEquation (4.8) for low-pressure data, the value of

$k_{0}^{i} = {3\frac{B_{i}}{A_{u}}}$can be determined. From Equation (4.3), the adsorption isotherm issimply estimated as:

$\begin{matrix}{{{\Delta\; H^{i}}❘\left\langle q \right\rangle_{i}} = {{{\Delta\; H^{i}\frac{A_{i}}{3B_{i}}\frac{d\left\langle n \right\rangle_{i}}{d\; t}} + {\Delta\; H^{i}\left\langle n \right\rangle_{i}}} = {\frac{A_{i}}{3B_{i}}e^{\frac{3B_{i}}{2A_{i}}t}{{\frac{d}{d\; t}\left\lbrack {\Delta\; H^{i}\left\langle n \right\rangle_{i}e^{3\frac{B_{i_{t}}}{A_{i}}}} \right\rbrack}.}}}} & (4.9)\end{matrix}$

Finally, one can measure the heat of adsorption by performing a separateexperiment at a different initial temperature and calculate the value ofΔH^(i)

q

_(i). One way to estimate ΔH^(i) is to exploit the standardthermodynamic relation:

$\begin{matrix}{{\Delta\; H^{i}} = {{- \frac{R_{g}T^{2}}{p}}{\frac{{\partial q_{i}}/{\partial T_{i}}}{{\partial q_{i}}/{\partial p}}.}}} & (4.10)\end{matrix}$

If the dependence of the isosteric heat of adsorption, ΔH^(i), ontemperature and pressure is neglected, Equation 4.10 can be written as:

$\begin{matrix}{{\Delta\; H^{i}} = {{- \frac{R_{g}T^{2}}{p}}{\frac{{\partial q_{i}}/{\partial T_{t}}}{{\partial q_{i}}/{\partial p}}.}}} & (4.11)\end{matrix}$

Thus, one can directly use Equation 4.11 to estimate the heat ofadsorption of tested materials when the values of ΔH^(i)qi is estimatedat different pressures and temperatures. Once the heat of adsorption isestimated, one can obtain high-resolution adsorption isotherms that canbe used to rank adsorbents.

FIG. 2 is a flowchart of a method for using the apparatus to screensorbents. In light of Equations 4.5, 4.8, and 4.11, the screening method200 can be performed by following the experiment herein at threedifferent conditions. At block 202, in the first experiment, the initialchamber temperature, T_(ini), is set equal to the cooling fluidtemperature set at T_(f1) and the cooling fluid velocity is set at aparticular value such that the value of the overall heat transfercoefficient of the device is equal to h_(f1). At block 204, in thesecond experiment, the initial temperature is also set at T_(f1), butthe fluid velocity is changed such as the value of heat transfercoefficient is different from the first experiment and is equal toh_(f2). At block 206, the third experiment is performed at a differentinitial temperature, that is equal to the fluid temperature T_(f2), andthe velocity of the cooling fluid is equal to that used in either thefirst or the second experiment. At block 208, the temperature profilesmeasured for all samples in the first two experiment is used todetermine the amount of generated heat, ΔH^(i)qi, at T_(f1) in Equations4.5 and 4.8. At block 210, the temperature transient profile in thethird experiment is used in Equations 4.5 and 4.6 to estimate the valueof the heat generated amount, ΔH^(i)q_(i), at T_(f2). At block 212, theheat generated data at the different fluid temperatures, T_(f1) andT_(f2), and Equation 4.11, is used to estimate the heat of adsorptionand obtain the adsorption isotherm, q_(i), at these temperatures.

EXAMPLES

Simulation of Experiments

The adsorption experiments were simulated using the COMSOL 5.4a packagein comparison to experimentally verified adsorption equations.

Example: Screening Materials for Methane Adsorption

FIG. 3A is a plot of methane adsorption isotherms of a set of selectedmaterials at 10° C. In this experiment, q′=q/(1−∈). In this example, thescreening method is illustrated by estimating the methane adsorptioncapacity of different porous materials from modeled adsorption dynamics.To assess the accuracy of the method, the results are compared with theexperimental adsorption isotherms. The experimental adsorption isothermsof these materials along with their thermal and physical properties arelisted in Table A.1 in Appendix A. The temperature variation in a givensample well is modeled. The model neglects temperature variations andpressure drop along the axial direction. It assumes that heat transferwithin the sample well is dominated by conduction through an effectivesolid medium. The gas phase is in thermal equilibrium with the solidadsorbent; the temperature of the gas phase and the solid particles areassumed to be equal. Assumptions used in this model have been verifiedin other studies through simulating a three dimensional case of naturalgas adsorption dynamics. Depending on the geometry of the adsorbent bed,different variants of the model are used to study the effects ofnon-isothermal adsorption dynamics in adsorbed gas storage applications.

When the thermal interference between the samples is ignored, due totheir large separation distance, assumptions used in Chang and Talumodel are applicable to the adsorption process that is taking place inthe described device. These assumptions include ignoring the temperaturegradient in the axial direction, since the aspect ratio of the samplewell, L/R, is much larger than 1. Furthermore, temperature variationsalong the angular position are neglected due to radial symmetry.Therefore, an energy balance over any sample well volume yields a onedimensional equation:

$\begin{matrix}{{{\left\lbrack {{\rho_{ɛ}{C_{p_{s}}\left( {1 - \epsilon} \right)}} + {{\epsilon\rho}_{g}C_{p_{S}}}} \right\rbrack\frac{\partial T}{\partial t}} = {{k_{eff}\frac{1}{r}\frac{\partial}{\partial r}\left( {r\frac{\partial T}{\partial r}} \right)} + {\Delta\; H\frac{\partial n}{\partial t}} + {\epsilon\frac{\partial p}{\partial t}}}},} & 5.1\end{matrix}$where ∈ is the porosity of the sample well accounting for voids inbetween adsorbent particles and the intrinsic porosity of the adsorbent.C_(pg) and C_(ps) are the heat capacities of the gas and adsorbent,respectively. ρ_(g) and ρ_(s) are the density of the gas and adsorbentsolid, respectively. k_(eff) is the effective thermal conductivitydefined as k_(eff)=[(1−∈)k_(s)−∈k_(g)]. ΔH is the isosteric heat ofadsorption that depends on pressure and temperature. Convection heattransfer at the wall of the sample well is given by:

$\begin{matrix}{{{{- k_{eff}}\frac{\partial T}{\partial r}} = {h_{f}\left( {T - T_{f}} \right)}},{{{at}\mspace{14mu} r} = {R.}}} & {5.2a}\end{matrix}$Due to cylindrical symmetry, the other boundary condition can be writtenas:

$\begin{matrix}{{\frac{\partial T}{\partial r} = 0},{{{at}\mspace{14mu} r} = 0.}} & {5.2b}\end{matrix}$The adsorption rate term, ∂n/∂t, is given by:

$\begin{matrix}{\frac{\partial n}{\partial t} = {{k_{0}\left( {q - n} \right)}.}} & 5.3\end{matrix}$

Finally, pressure drop across the sample cell is neglected and thussolving a momentum balance is redundant. The mass conservation over thesample well is written as:

$\begin{matrix}{{{{\epsilon\frac{\partial c}{\partial t}} + \frac{\partial n}{\partial t}} = {{\frac{K}{\mu\; r}\frac{\partial}{\partial r}\left( {r\frac{\partial p}{\partial r}} \right)} + \overset{.}{f}}},} & 5.4\end{matrix}$where c is the molar concentration of the free gas and for an ideal gas,it is given by c=p/R_(g)T and f is the molar flow rate to the cell pervolume of the medium. K is the permeability of the sample well and μ isthe test gas viscosity. The no flux condition at the sample wall isrequired to solve Equation 5.4 and is given by:

$\begin{matrix}{{\frac{\partial p}{\partial r} = 0},{{{at}\mspace{14mu} r} = {R.}}} & 5.5\end{matrix}$

In this example, the pressurization rate is controlled. Thus, Equation5.4 is decoupled from the energy balance and is useful to determine themass flow rate to the cell. Equations 5.1-5.5 are the general equationsgoverning heat transfer within the sample wells 123. By specifying thefunctional form of the equilibrium adsorption isotherm, q, one cansimulate the adsorption dynamics as methane is introduced to the cell.The functional form the adsorption isotherms of the tested materials arelisted in Table A.1.

FIG. 3B is a plot of the transient temperature profile of the selectedmaterials for simulation case I. In this example, methane is introducedto the sample cell and the pressurization rate is controlled at∂p/∂t=274 Pa s⁻¹. The radius of the sample wells 123 is set to be 0.6 mmto minimize the effects of heat transfer resistance of the adsorbentmaterials. As can be seen in FIG. 3B, the methane adsorption capacity ofMOF-5 is lower than that of MOF-74 but becomes higher at a higherpressure. Such crossover in the relative adsorption capacity is hard todetect when generating low-resolution adsorption isotherms.

FIG. 4A is a plot of the generated heat profile when the adsorptionexperiment is performed using a cooling fluid with a temperature of 10°C. Using Equations 5.1-5.4, the adsorption dynamics of CH₄ in thesematerials was simulated using the COMSOL 5.4a package at three differentconditions. Table 5.1 summarizes the parameters used in the simulation.

Even though the adsorption capacity of MOF-74, CUBTC, and AX-21 issimilar at low pressures, they exhibit different temperature increasesearly in the adsorption process. Thus, material ranking cannot not bebased on direct comparison of observed temperature profiles.Furthermore, temperature increase for some materials may be lower thanthe detection resolution of the used infrared camera. It is advisable touse a low enough fluid velocity such that a temperature increase in allporous materials can be detected while maintaining an overalltemperature increase of most materials that is much smaller than thecooling fluid temperature. Finally, the observed delay in the peak ofthe temperature of material MIL-101 is due to the significantly lowerthermal conductivity of this material compared to others. Theconductivity of MIL-101 is equal to 0.05 W/m K while the conductivity ofthe other materials is of the order of 0.3 W/m K. Since the thermalresistance of MIL-101 is significant, inaccuracies in the estimate ofthe adsorption isotherm may arise. To avoid such inaccuracies, the radiiof the sample wells 123 can be decreased.

TABLE 5.1 Simulation conditions used to simulate the adsorption dynamicsof the materials described in Table A.1 Simulation: I Simulation: IISimulation: III T(0) = Tf1 = 10° C. T(0) = Tf1 = 10° C. T(0) = Tf1 = 25°C. hf1 = 50 W/(m2 K) hf1 = 100 W/(m2 K) hf1 = 50 W/(m2 K) R = 0.6 mm kg= 0.03 W/m K ∂p/∂t = 274 Pa s⁻¹ C. pg = 2.23 kJ/kg K

FIG. 4B is a plot of the calculated value of the generated heat ofadsorption of the selected materials over time. Following the screeningmethod described herein, the temperature profiles from cases I and IIare used to calculate the effective heat capacity, given by Equation4.6, and thus estimating the generated heat, ΔH

n

, of the samples for simulation cases I and III. To calculate the masstransfer coefficient, k₀, of all samples, data of the generated heat, atearly times, shown in FIG. 4B, are fitted using Equation 4.8. Once themass transfer coefficient of a given sample is determined, thevolumetric heat of adsorption (i.e. per volume of the adsorbent), ΔH

q

, can be directly calculated using Equation 4.9. Using Equation 4.11 anddata of ΔH

q

at 10° C. and 25° C., the isosteric heat of adsorption, ΔH, iscalculated as a function of the volumetric heat of adsorption.

FIG. 5 is a plot of the calculated heat of adsorption of the selectedmaterials using Equation 5.11. As shown in FIG. 5 , the assumption ofconstant heat of adsorption is violated in the low and high-pressureregimes. Therefore, an average value in the medium pressure region wherethe heat of adsorption does not vary much is used.

FIGS. 6A and 6B are plots of estimated adsorption isotherms for theselected materials and a comparison with actual values. Once the heat ofadsorption is determined, high-resolution adsorption isotherms aregenerated as shown in FIG. 6A, which shows the measured adsorptionisotherms at 10° C. The crossover between the adsorption capacity ofMOF-5 and MOF-74 at high pressure is retrieved. To assess the accuracyof the screening method presented in this invention, the estimatedadsorption amount, (q), is directly compared with the experimentalvalues modeled using the isotherms presented in Table A.1. FIG. 6B showsa comparison between the estimated and actual adsorbed amount at T1=10°C. and T2=25° C. The solid black line is a 45-degree line that one wouldget if the estimated adsorbed amount is exactly equal to that presentedin Table A.1. FIG. 6B indicates that the presented screening method isvery accurate at low pressures but consistently underestimates theadsorption capacity at high pressures. The errors introduced inestimating the adsorption amount is very sensitive to the calculationsof the heat of adsorption. For screening purposes, such errors aretolerable as one is only interested in the relative adsorption capacitybetween different samples at different pressures. Moreover, errors inestimating the adsorption capacity of MIL-101 are the largest amongstsimulated materials. This is due to the anomalously higher resistancefor heat transfer such that the Biot number of the sample interior isequal to 4 while derivations for Equation 4.5 assume that the value ofBis<<1.

FIG. 7 is a plot of model adsorption isotherms of the selectedmaterials. The modelling was performed at 10° C. using equations listedin Table A.2. In the model, q′=q/(p_(s)(1−ε)).

Example: Screening Materials for Carbon Dioxide Adsorption

FIG. 8 is a plot of temperature profiles during adsorption of CO₂. Theresults shown are for simulation case I. Similar to the previousexample, CO₂ adsorption dynamics can be simulated by specifying thefunctional form of the equilibrium adsorption isotherm, q, and otherintrinsic physical and thermal properties of the material (Table A.2).The parameters for the simulation cases are the same as those listedTable 5.1. The thermal properties of CO₂ used in the simulation areCpg=0.9 kJ/kg K and kg=0.017 W/m K. As shown in FIG. 8 , differences inthe isotherm types and shapes, e.g., adsorption rate, saturationpressure, and saturation capacity, again confirm the need forhigh-throughput methods that provide HRIs for one to unambiguouslyscreen and rank the adsorption capacity of the selected sorbentmaterials.

As shown in FIG. 8 , while MOF-74 does not possess the highestadsorption capacity of the selected materials (FIG. 7 ), its temperaturepeak is an order of magnitude higher than the other materials due to thesteep adsorption isotherm at low pressure (i.e. high dq/dp) and therelatively high heat of adsorption (Table A-2). Following the sameprocedure outlined in the previous section, the heat of adsorption, ΔH,was calculated and plotted in FIG. 9 .

FIG. 9 is a plot of the estimated heats of adsorption of CO₂ for theselected materials. As shown in FIG. 9 , the assumption of constant heatof adsorption is violated at low and high-pressure regimes. Therefore,an average value in the medium pressure regime where the heat ofadsorption is largely constant was used.

FIGS. 10A and 10B are plots of the estimated adsorption isotherms forselected materials and their comparisons with actual values. Once theheat of adsorption is determined, the high-resolution adsorptionisotherms are generated as shown in FIG. 10A. Except for MOF-74, thecrossover between the adsorption capacity of MOF-177, CUBTC, and USO2Niat high pressure is retrieved. MOF-74 surface heterogeneity introducelarge variations in the estimated isostatic heat of adsorption

ΔH

obtained from the

ΔH n

isotherm rendering the constant ΔH assumption and hence the method ofcalculating ΔH (Equation (4.11)) invalid to estimate the adsorptioncapacity,

q

at high pressure. To assess the accuracy of the screening methodpresented in this invention, the estimated adsorption amount,

q

, is directly compared with the experimental values modeled using theisotherms presented in Table A-2.

FIG. 10B shows the estimated and actual adsorbed amount at differentpressure and temperature. The solid black line is a 45-degree line thatone would get if the estimated adsorbed amount were exactly equal tothat presented in Table A-2. FIG. 10B indicates that the presentedscreening method is very accurate at low pressures but consistentlyunderestimates the adsorption capacity at high pressures. The errorsintroduced in estimating the adsorption amount is very sensitive to thecalculations of the heat of adsorption.

As mentioned previously, the adsorption capacity of MOF-74 deviatessignificantly from the experimental values due to the large variationsof ΔH because of the significant surface heterogeneity of the material.One may consider detailed analysis of materials that show significantdeviation from the constant ΔH assumption when estimating theequilibrium adsorption isotherm. For screening purposes, such errors aretolerable as one is interested in the relative adsorption capacitybetween different samples at different pressures but not the actual.

Physical properties of sorbents used in Examples 1 and 2.

Table A.1 lists the physical properties of the sorbents used to simulatemethane adsorption while Table A.2 lists the physical properties used tosimulate CO₂ adsorption. Parameters that were not found in theliterature were estimated. The estimated properties did not change theperformed assessment.

TABLE A.1 CH₄ adsorption isotherms and sorbents intrinsic propertiesIsotherm Parameter value CUBTC Linker: benzene- 1,3,5-tricarboxylateMetal: Cupper $\begin{matrix}{\frac{q}{p_{s}q_{m}} = \frac{bP}{{bP} + 1}} \\{b = {b_{0}e^{\frac{\Delta\; H}{R_{g}T}}}}\end{matrix}\quad$ q_(m) = 15.9 (mol/kg) b0 = 0.834 (GPa⁻¹) ΔH = 16.55(kJ/mol) k₀ = 0.15 (s⁻¹)* ρ_(s) = 703 (kg/m³) k_(s) = 0.39 (W/m K)C_(ps) = 777.21 (J/kg K) ϵ = 0.5** MOF-5 Linker: Terephthalic acidMetal: Zinc $\begin{matrix}{\frac{q}{p_{s}q_{m}} = \frac{bP}{{bP} + 1}} \\{b = {b_{0}e^{\frac{\Delta\; H}{R_{g}T}}}}\end{matrix}\quad$ q_(m) = 30.5 (mol/kg) b0 = 1.01 (GPa⁻¹) ΔH = 12.3(kJ/mol) k₀ = 3.3 (s⁻¹) ρ_(s) = 621 (kg/m³) k_(s) = 0.32 (W/m K) C_(ps)= 750 (J/kg K) ϵ = 0.5** MIL-101 Linker: Terephthalic acid Metal:Chromium Salts $\begin{matrix}{\frac{q}{p_{s}q_{m}} = \frac{bP}{{bP} + 1}} \\{b = {b_{0}e^{\frac{\Delta\; H}{R_{g}T}}}}\end{matrix}\quad$ q_(m) = 34 (mol/kg) b₀ = 1.79 (GPa⁻¹) ΔH = 9.9(kJ/mol) k₀ = 0.025 (s⁻¹)*** ρ_(s) = 440 (kg/m³) k_(s) = 0.05 (W/m K)C_(ps) = 643 (J/kg K) ϵ = 0.5** MOF-74 Linker: 2,5-Dihydroxyterephthalic acid Metal: Magnesium $\begin{matrix}{\frac{q}{\rho_{s}} = {\sum\limits_{i = 1}^{2}\frac{q_{m_{i}}b_{i}P}{{b_{i}P} + 1}}} \\{{\Delta\; H} = \frac{{\Delta\; H_{1}\beta_{1}} + {\Delta\; H_{2}\beta_{2}}}{\left( {\beta_{1} + \beta_{2}} \right)}} \\{\beta_{1{(2)}} = {q_{m_{1{(2)}}}{b_{1{(2)}}\left( {1 + {b_{2{(1)}}p}} \right)}^{2}}} \\{b_{i} = {b_{0_{i}}e^{\frac{\Delta\; H_{i}}{R_{g}T}}}}\end{matrix}\quad$ q_(m) ₁ = 11.0 (mol/kg) q_(m) ₂ = 5.0 (mol/kg) b₀₁ =3.01 × 10−4 (MPa⁻¹) b₀₂ = 4.08 × 10−5 (MPa⁻¹) ΔH₁ = 20.5 (kJ/mol) ΔH₁ =16.0 (kJ/mol) k₀ = 0.6 (s⁻¹) ρ_(s) = 911 (kg/m³) k_(s) = 0.3 (W/m K)C_(ps) = 900 (J/kg K) ϵ = 0.5** AX-21 Activated carbon $\begin{matrix}{\frac{q}{\rho_{s}} = {\sum\limits_{i = 1}^{2}\frac{q_{m_{i}}b_{i}P}{{b_{i}P} + 1}}} \\{{\Delta\; H} = \frac{{\Delta\; H_{1}\beta_{1}} + {\Delta\; H_{2}\beta_{2}}}{\left( {\beta_{1} + \beta_{2}} \right)}} \\{\beta_{1} = {q_{m_{i}}{b_{i}\left( {1 + {b_{i}p}} \right)}^{2}}} \\{b_{i} = {b_{0_{i}}e^{\frac{\Delta\; H_{i}}{R_{g}T}}}}\end{matrix}\quad$ q_(m) ₁ = 28.3 (mol/kg) q_(m) ₂ = 10.5 (mol/kg) b₀₁ =1.01 (GPa⁻¹) b₀₂ = 1.23 (GPa⁻¹) ΔH₁ = 10.7 (kJ/mol) ΔH₂ = 16.6 (kJ/mol)k₀ = 0.1215 (s⁻¹)*** ρ_(s) = 972 (kg/m³) k_(s) = 0.15 (W/m K) C_(ps) =844 (J/kg K) ϵ = 0.5** *Estimated based on diffusion coefficient, D, of1 × 10−10 m/s [26] and MOF-5 and MOF-177 particle radius, rc, of 1 ×10−4 m [29] where k₀ = 15D/rc2. **For simplicity, porosity, ϵ, isassumed to be 0.5 for all materials with no impact to the results.***Due to lack of published values, an estimated mass transfercoefficient, k₀, based on similar materials, is used noting that suchchoice does not affect the equilibrium adsorption results.

TABLE A.2 CO₂ adsorption isotherms and sorbents intrinsic propertiesIsotherm Parameter value CUBTC Linker: benzene- 1,3,5-tricarboxylateMetal: Cupper $\begin{matrix}{\frac{q}{\rho_{s}q_{m}} = \frac{bP}{{bP} + 1}} \\{b = {b_{0}e^{\frac{\Delta\; H}{R_{g}T}}}}\end{matrix}\quad$ q_(m) = 18.2 (mol/kg) b₀ = 1.37 × 10−4 (MPa⁻¹) ΔH =25.5 (kJ/mol) k₀ = 0.23 (s⁻¹) ρ_(s) = 703 (kg/m³) k_(s) = 0.39 (W/m K)C_(ps) = 777.21 (J/kg K) ϵ = 0.5* MOF-177 Linker: 1,3,5-tris(4-carboxyphenyl) benzene Metal: Zinc salt $\begin{matrix}{\frac{q}{\rho_{s}q_{m}} = \frac{bP}{{bP} + 1}} \\{b = {b_{0}e^{\frac{\Delta\; H}{R_{g}T}}}}\end{matrix}\quad$ q_(m) = 48.0 (mol/kg) b₀ = 8.06 × 10−4 (MPa⁻¹) ΔH =14.0 (kJ/mol) k₀ = 0.1597 (s⁻¹) ρ_(s) = 477 (kg/m³) k_(s) = 0.3 (W/m K)C_(ps) = 490 (J/kg K) ϵ = 0.5* USO-2-Ni Ni2(1,4- bdc)2(dabco) · 4D MF ·0.5H2O $\begin{matrix}{\frac{q}{\rho_{s}q_{m}} = \frac{{\alpha\; P} + {2\beta\; P^{2}}}{1 + {\alpha\; P} + {\beta\; P^{2}}}} \\{q_{m} = {a\mspace{14mu}{\exp\left\lbrack {{{- b}/R_{g}}T} \right\rbrack}}} \\{{\alpha = {A\mspace{14mu}{\exp\left\lbrack {{{- B}/R_{g}}T} \right\rbrack}}}{\beta = {F\mspace{14mu}{\exp\left\lbrack {{{- G}/R_{g}}T} \right\rbrack}}}}\end{matrix}{\quad\quad}$ a = 2.1 (mol/kg) A = 2.61 (GPa⁻¹) F = 1.03(GPa−2) b = −2.96 (kJ/mol) B = −16.75 (kJ/mol) G = −38.61 (kJ/mol) ΔH =19.9 (kJ/mol) k₀ = 0.5 (s⁻¹) ρ_(s) = 531 (kg/m³) k_(s) = 0.35 (W/m K)C_(ps) = 1160 (J/kg K) ϵ = 0.5* MOF-74 Linker: 2,5-Dihydroxyterephthalic acid Metal: Magnesium $\begin{matrix}{\frac{q}{\rho_{s}} = {\sum\limits_{i = 1}^{2}\frac{q_{m_{i}}b_{i}P}{{b_{i}P} + 1}}} \\{{\Delta\; H} = \frac{{\Delta\; H_{1}\beta_{1}} + {\Delta\; H_{2}\beta_{2}}}{\left( {\beta_{1} + \beta_{2}} \right)}} \\{\beta_{1{(2)}} = {q_{m_{1{(2)}}}{b_{1{(2)}}\left( {1 + {b_{2{(1)}}p}} \right.}}} \\{b_{i} = {b_{0_{i}}e^{\frac{\Delta\; H_{i}}{R_{g}T}}}}\end{matrix}{\quad\quad}$ q_(m) ₁ = 6.8 (mol/kg) q_(m) ₂ = 9.9 (mol/kg)b₀₁ = 2.44 × 10−5 (MPa⁻¹) b₀₂ = 1.39 × 10−4 (MPa⁻¹) ΔH₁ = 42.0 (kJ/mol)ΔH₂ = 24.0 (kJ/mol) k₀ = 0.1215 (s⁻¹) ρ_(s) = 911 (kg/m³) k_(s) = 0.3(W/m K) C_(ps) = 900 (J/kg K) ϵ = 0.5* UIO-67 Zr based MOF$\begin{matrix}{\frac{q}{\rho_{s}q_{m}} = \frac{bP}{1 + {bP}}} \\{q_{m} = {q^{\infty}\mspace{14mu}{\exp\left\lbrack {{{- \alpha}/R_{g}}T} \right\rbrack}}} \\{{\alpha = {A\mspace{14mu}{\exp\left\lbrack {{{- B}/R_{g}}T} \right\rbrack}}}{b = {\gamma\mspace{14mu}{\exp\left\lbrack {{{- \beta}/R_{g}}T} \right\rbrack}}}}\end{matrix}{\quad\quad}$ q^(∞) = 2.3 (mol/kg) γ = 2.82 (GPa⁻¹) α =−5.02 (kJ/mol) β = −12.51 (kJ/mol) ΔH = 19.0 (kJ/mol) k₀ = 0.5 (s⁻¹)ρ_(s) = 557 (kg/m³) k_(s) = 0.35 (W/m K) C_(ps) = 1250 (J/kg K) ϵ = 0.5**For simplicity, porosity, c, is assumed to be 0.5 for all materialswith no affect on the results.

Examples described herein include the following embodiments,

An embodiment described herein provides a system for screening sorbents.The system includes a sample chamber with a hermetic seal and a heatexchanger system. The heat exchanger system includes a heat exchangerdisposed in the sample chamber, a coolant circulator fluidically coupledto the heat exchanger, and a sample plate comprising sample wells 123 incontact with the cooling fluid from the coolant circulator. The systemalso includes a gas delivery system. The gas delivery system includes agas source and a flow regulator. A temperature measurement system isconfigured to sense the temperature of the sample wells 123.

In an aspect, the system includes an instrumentation control systemcoupled to the temperature measurement system to monitor the temperatureof the sample wells 123 over time. In an aspect, the system includes apressure sensor on the sample chamber, wherein the pressure sensor iscoupled to the instrumentation control system.

In an aspect, the system includes a vacuum pump coupled to the samplechamber. In an aspect, the system includes a vent valve coupled to thesample chamber.

In an aspect, the sample chamber includes a lid, wherein the lidincludes an infrared-transparent window disposed over the sample wells123. In an aspect, the infrared-transparent window includes quartz orglass.

In an aspect, the temperature measurement system includes an infraredcamera. In an aspect, the infrared camera collects data at least at 60frames per second.

In an aspect, the sample wells 123 are less than 2 mm in inner diameter.In an aspect, the sample wells 123 are about 30 mm in depth. In anaspect, the sample wells 123 are arranged in a staggered configurationon the sample plate.

Another embodiment described herein provides a method for screeningsorbents. The method includes measuring temperatures of adsorption forsorbents at a first condition, measuring the temperatures of adsorptionof the sorbents at a second condition, and calculating generated heatfrom the measurements at the first condition and second condition. Themethod includes measuring the temperatures of adsorption of the sorbentsat a third condition, calculating a temperature transient profile fromthe measurements at the third condition, and calculating a heat ofadsorption from measurements collected at two temperatures. Further, themethod includes calculating an adsorption isotherm from measurementscollected at two temperatures.

In an aspect, the method includes loading sorbents samples into samplewells 123 in a sample plate in a heat exchanger, sealing a lid over asample chamber holding the heat exchanger, and pulling a vacuum on thesample chamber.

In an aspect, the method includes controlling a temperature of the heatexchanger at a temperature of interest, adding a test gas of interestcontinuously, and monitoring the temperature of the samples.

In an aspect, the method includes venting the test gas, and pulling avacuum on the sample chamber to start a new test.

In an aspect, the method includes setting the first condition to be atan initial temperature (T_(ini)) of T_(f1) and a heat transfercoefficient of h_(f1). In an aspect, the method includes setting thesecond condition to be at an initial temperature (T_(ini)) of T_(f1) anda heat transfer coefficient of h_(f2). In an aspect, the method includessetting the third condition to be at a heat transfer coefficient ofh_(f1).

In an aspect, the method includes calculating the generated heat fromthe measurements at the first condition and the second condition usingthe equation:

${\Delta\; H^{i}\left\langle n \right\rangle_{i}} = {\frac{1}{R}\left\{ {{h_{f}^{1}{\int_{0}^{p}{\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{\left( \frac{dp}{dt} \right)}{dp}}}} + {\frac{\Delta\;\left\langle T \right\rangle_{i_{1}}}{{\Delta\left\langle T \right\rangle_{i_{2}}\left( p_{z} \right)} - {\Delta\left\langle T \right\rangle_{i_{2}}\left( v_{2} \right)}}\left. \quad\left\lbrack {{h_{f}^{2}{\int_{0}^{p_{s}}{\frac{\Delta\left\langle T \right\rangle_{i_{2}}}{\left( \frac{dp}{dt} \right)}{dp}}}} - {h_{f}^{1}{\int_{0}^{p_{s}}{\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{\left( \frac{dp}{dt} \right)}{dp}}}}} \right\rbrack \right\}} - {\epsilon\;{p.}}} \right.}$

In an aspect, the method includes calculating the generated heat fromthe measurements at the first condition and the second condition usingthe equation:

${\Delta\; H^{i}\left\langle n \right\rangle_{i}} = {{{\frac{\Delta\; H^{i}C_{i}k_{0}^{i}}{2}t^{2}} - {\frac{\Delta\; H^{i}C_{i}k_{0}^{i^{2}}}{6}t^{3}} + {O\left( t^{4} \right)}} = {{A_{i}t^{2}} - {B_{i}{t^{3}.}}}}$

In an aspect, the method includes calculating heat of adsorption fromdata measured at two different temperatures using the equation:

${\Delta\; H^{i}} = {{- \frac{R_{g}T^{2}}{p}}{\frac{{\partial q_{i}}/{\partial T_{i}}}{{\partial q_{i}}/{\partial p}}.}}$Other implementations are also within the scope of the following claims.

What is claimed is:
 1. A method for screening sorbents, comprising:measuring temperatures of adsorption for sorbents at a first condition;measuring the temperatures of adsorption of the sorbents at a secondcondition; calculating generated heat from the measurements at the firstcondition and second condition; measuring the temperatures of adsorptionof the sorbents at a third condition; calculating a temperaturetransient profile from the measurements at the third condition; andcalculating a heat of adsorption from measurements collected at twotemperatures; and calculating an adsorption isotherm from measurementscollected at two temperatures.
 2. The method of claim 1, comprising:loading sorbent samples into sample wells in a sample plate in a heatexchanger; sealing a lid over a sample chamber holding the heatexchanger; and pulling a vacuum on the sample chamber.
 3. The method ofclaim 2, comprising: controlling a temperature of the heat exchanger ata temperature of interest; adding a test gas of interest continuously;and monitoring the temperature of the sorbent samples.
 4. The method ofclaim 3, comprising: venting the test gas; and pulling a vacuum on thesample chamber to start a new test.
 5. The method of claim 1, comprisingsetting the first condition to be at an initial temperature (T_(ini)) ofT_(f1) and a heat transfer coefficient of h_(f1).
 6. The method of claim1, comprising setting the second condition to be at an initialtemperature (T_(ini)) of T_(f1) and a heat transfer coefficient ofh_(f2).
 7. The method of claim 1, comprising setting the third conditionto be at a heat transfer coefficient of h_(f1).
 8. The method of claim1, comprising calculating the generated heat from the measurements atthe first condition and the second condition using the equation:${\Delta\; H^{i}\left\langle n \right\rangle_{i}} = {\frac{1}{R}\left\{ {{h_{f}^{1}{\int_{0}^{v}{\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{\left( \frac{dp}{dt} \right)}{dp}}}} + {\frac{\Delta\;\left\langle T \right\rangle_{i_{1}}}{{\Delta\left\langle T \right\rangle_{i_{2}}\left( p_{z} \right)} - {\Delta\left\langle T \right\rangle_{i_{2}}\left( p_{2} \right)}}\left. \quad\left\lbrack {{h_{f}^{2}{\int_{0}^{p_{s}}{\frac{\Delta\left\langle T \right\rangle_{i_{2}}}{\left( \frac{dp}{dt} \right)}{dp}}}} - {h_{f}^{1}{\int_{0}^{p_{s}}{\frac{\Delta\left\langle T \right\rangle_{i_{1}}}{\left( \frac{dp}{dt} \right)}{dp}}}}} \right\rbrack \right\}} - {\epsilon\;{p.}}} \right.}$9. The method of claim 1, comprising calculating the generated heat fromthe measurements at the first condition and the second condition usingthe equation:${\Delta\; H^{i}\left\langle n \right\rangle_{i}} = {{{\frac{\Delta\; H^{i}C_{i}k_{0}^{i}}{2}t^{2}} - {\frac{\Delta\; H^{i}C_{i}k_{0}^{i^{2}}}{6}t^{3}} + {O\left( t^{4} \right)}} = {{A_{i}t^{2}} - {B_{i}{t^{3}.}}}}$10. The method of claim 1, comprising calculating heat of adsorptionfrom data measured at two different temperatures using the equation:${\Delta\; H^{i}} = {{- \frac{R_{g}T^{2}}{p}}{\frac{{\partial q_{i}}/{\partial T_{i}}}{{\partial q_{i}}/{\partial p}}.}}$